The Kantorovich norm is a mathematical concept that measures the distance between two probability measures. It is named after the Russian mathematician Leonid Kantorovich and is commonly used in the field of optimal transport.
The Kantorovich norm is calculated by finding the supremum (or maximum) of the difference between two integrals, where one integral is taken over all possible transport plans between the two probability measures. This supremum is then divided by the total mass that needs to be transported.
The pushforward measure is a way to transform one probability measure into another by applying a function. In the context of the Kantorovich norm, the pushforward measure is used to map one probability measure onto another, allowing for the calculation of the distance between the two measures.
The Kantorovich norm is used in a variety of applications, including image processing, economics, and machine learning. One common use is in transportation planning, where it can help optimize the flow of goods or people from one location to another.
The Kantorovich norm is significant in mathematics because it provides a way to measure the distance between two probability measures, which is a fundamental concept in many fields of mathematics. It has also been used in the development of new mathematical theories and applications, making it an important tool for researchers and practitioners alike.